Subspace of Either-Or Space less Zero is not Lindelöf

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $T = \left({S, \tau}\right)$ be the either-or space.

Let $H = S \setminus \left\{{0}\right\}$ be the set $S$ without zero.


Then the topological subspace $T_H = \left({H, \tau_H}\right)$ is not a Lindelöf space.


Proof

By definition of topological subspace, $U \subseteq H$ is open in $T_H$ if and only if:

$(1): \quad \left\{{0}\right\} \nsubseteq U$

or:

$(2): \quad \left({-1 \,.\,.\, 1}\right) \subseteq U$

But for all $U \subseteq H$, condition $(1)$ holds as $0 \notin H$.

So $T_H$ is by definition a discrete space.

As $T_H$ is uncountable, we have that Uncountable Discrete Space is not Lindelöf holds.

Hence the result.

$\blacksquare$


Sources